Mittag--Leffler Euler integrator and large deviations for stochastic space-time fractional diffusion equations

Stochastic space-time fractional diffusion equations often appear in the modeling of the heat propagation in non-homogeneous medium. In this paper, we firstly investigate the Mittag--Leffler Euler integrator of a class of stochastic space-time fractional diffusion equations, whose super-convergence order is obtained by developing a helpful decomposition way for the time-fractional integral. Here, the developed decomposition way is the key to dealing with the singularity of the solution operator. Moreover, we study the Freidlin--Wentzell type large deviation principles of the underlying equation and its Mittag--Leffler Euler integrator based on the weak convergence approach. In particular, we prove that the large deviation rate function of the Mittag--Leffler Euler integrator $\Gamma$-converges to that of the underlying equation

Numerical approximations of one-point large deviations rate functions of stochastic differential equations with small noise

In this paper, we study the numerical approximation of the one-point large deviations rate functions of nonlinear stochastic differential equations (SDEs) with small noise. We show that the stochastic $\theta$-method satisfies the one-point large deviations principle with a discrete rate function for sufficiently small step-size, and present a uniform error estimate between the discrete rate function and the continuous one on bounded sets in terms of step-size. It is proved that the convergence orders in the cases of multiplicative noises and additive noises are $1/2$ and $1$ respectively. Based on the above results, we obtain an effective approach to numerically approximating the large deviations rate functions of nonlinear SDEs with small time. To the best of our knowledge, this is the first result on the convergence rate of discrete rate functions for approximating the one-point large deviations rate functions associated with nonlinear SDEs with small noise

Error analysis of numerical methods on graded meshes for stochastic Volterra equations

This paper presents the error analysis of numerical methods on graded meshes for stochastic Volterra equations with weakly singular kernels. We first prove a novel regularity estimate for the exact solution via analyzing the associated convolution structure. This reveals that the exact solution exhibits an initial singularity in the sense that its H\"older continuous exponent on any neighborhood of $t=0$ is lower than that on every compact subset of $(0,T]$. Motivated by the initial singularity, we then construct the Euler--Maruyama method, fast Euler--Maruyama method, and Milstein method based on graded meshes. By establishing their pointwise-in-time error estimates, we give the grading exponents of meshes to attain the optimal uniform-in-time convergence orders, where the convergence orders improve those of the uniform mesh case. Numerical experiments are finally reported to confirm the sharpness of theoretical findings

A splitting semi-implicit method for stochastic incompressible Euler equations on T2\mathbb T^2

The main difficulty in studying numerical method for stochastic evolution equations (SEEs) lies in the treatment of the time discretization (J. Printems. [ESAIM Math. Model. Numer. Anal. (2001)]). Although fruitful results on numerical approximations for SEEs have been developed, as far as we know, none of them include that of stochastic incompressible Euler equations. To bridge this gap, this paper proposes and analyses a splitting semi-implicit method in temporal direction for stochastic incompressible Euler equations on torus $\mathbb{T}^2$ driven by an additive noise. By a Galerkin approximation and the fixed point technique, we establish the unique solvability of the proposed method. Based on the regularity estimates of both exact and numerical solutions, we measure the error in $L^2(\mathbb{T}^2)$ and show that the pathwise convergence order is nearly $\frac{1}{2}$ and the convergence order in probability is almost $1$

Convergence analysis of one-point large deviations rate functions of numerical discretizations for stochastic wave equations with small noise

In this work, we present the convergence analysis of one-point large deviations rate functions (LDRFs) of the spatial finite difference method (FDM) for stochastic wave equations with small noise, which is essentially about the asymptotical limit of minimization problems and not a trivial task for the nonlinear cases. In order to overcome the difficulty that objective functions for the original equation and the spatial FDM have different effective domains, we propose a new technical route for analyzing the pointwise convergence of the one-point LDRFs of the spatial FDM, based on the $\Gamma$-convergence of objective functions. Based on the new technical route, the intractable convergence analysis of one-point LDRFs boils down to the qualitative analysis of skeleton equations of the original equation and its numerical discretizations

Convergence of Density Approximations for Stochastic Heat Equation

This paper investigates the convergence of density approximations for stochastic heat equation in both uniform convergence topology and total variation distance. The convergence order of the densities in uniform convergence topology is shown to be exactly $1/2$ in the nonlinear case and nearly $1$ in the linear case. This result implies that the distributions of the approximations always converge to the distribution of the origin equation in total variation distance. As far as we know, this is the first result on the convergence of density approximations to the stochastic partial differential equation

Android HIV: A Study of Repackaging Malware for Evading Machine-Learning Detection

Machine learning based solutions have been successfully employed for automatic detection of malware in Android applications. However, machine learning models are known to lack robustness against inputs crafted by an adversary. So far, the adversarial examples can only deceive Android malware detectors that rely on syntactic features, and the perturbations can only be implemented by simply modifying Android manifest. While recent Android malware detectors rely more on semantic features from Dalvik bytecode rather than manifest, existing attacking/defending methods are no longer effective. In this paper, we introduce a new highly-effective attack that generates adversarial examples of Android malware and evades being detected by the current models. To this end, we propose a method of applying optimal perturbations onto Android APK using a substitute model. Based on the transferability concept, the perturbations that successfully deceive the substitute model are likely to deceive the original models as well. We develop an automated tool to generate the adversarial examples without human intervention to apply the attacks. In contrast to existing works, the adversarial examples crafted by our method can also deceive recent machine learning based detectors that rely on semantic features such as control-flow-graph. The perturbations can also be implemented directly onto APK's Dalvik bytecode rather than Android manifest to evade from recent detectors. We evaluated the proposed manipulation methods for adversarial examples by using the same datasets that Drebin and MaMadroid (5879 malware samples) used. Our results show that, the malware detection rates decreased from 96% to 1% in MaMaDroid, and from 97% to 1% in Drebin, with just a small distortion generated by our adversarial examples manipulation method.Comment: 15 pages, 11 figure